with \(\sum _{i}{r^1_i} = \sum _{j}{r^1_j} = 1,\;r^1_j,r^0_i\geqslant 0\)  and \(|\psi _{j}\rangle ,|\varphi _i\rangle \in \mathcal {H},\; i,j=1,2,\dots ,d\) . We assume that \(\lbrace |\psi _{j}\rangle \rbrace ,\lbrace |\varphi _i\rangle \rbrace \)  span the Hilbert space \(\mathcal {H}\)  of dimension \(d\) . Notice that we do not assume that \(\rho _0,\rho _1\)  commute; i.e. \(\lbrace |\psi _{j}\rangle \rbrace ^d_{j=1},\lbrace |\varphi _i\rangle \rbrace ^d_{i=1}\)  may be two different bases of \(\mathcal {H}\) . The target system produces quantum states \(|\varphi \rangle \in \mathcal {H}\)  which are collected by a detector (He), who wants to arrive at a decision rule \(\delta \in \mathcal {H}\rightarrow [0,1]\)  on which is the genuine characterization of the target system in (REF ): \(\delta (\varphi ) = k,\;k\in \lbrace 0,1\rbrace ,\)  when he thinks the hypothesis \(H_k\)  holds true. According to the measurement postulate of quantum mechanics [1]}, the detector decides by applying a measurement \(\Pi _1\in \mathcal {V}\)  upon the received quantum state \(|\varphi \rangle \) : \(\delta (\varphi ) = \langle \varphi |\Pi _1 |\varphi \rangle \) .
The decision rule \(\delta \) , or equivalently the measurement \(\Pi _1\) , leads to a detection rate \(P_D: \mathcal {V}\rightarrow [0,1]\)  and a false alarm rate \(P_F:\mathcal {V}\rightarrow [0,1]\)  as follows:
\(P_D(\Pi _1) = \text{Tr}(\Pi _1\rho _1), \;\; P_F(\Pi _1) = \text{Tr}(\Pi _1\rho _0).\) 
